Related papers: Dual Ramanujan-Fourier series
We give several expansion and identities involving the Ramanujan function $A_q$ and the Stieltjes--Wigert polynomials. Special values of our idenitities give $m$-versions of some of the items on the Slater list of Rogers-Ramanujan type…
On page 3 of his lost notebook, Ramanujan defines the Appell-Lerch sum $$\phi(q):=\sum_{n=0}^\infty \dfrac{(-q;q)_{2n}q^{n+1}}{(q;q^2)_{n+1}^2},$$ which is connected to some of his sixth order mock theta functions. Let $\sum_{n=1}^\infty…
We continue our study of convolution sums of two arithmetical functions $f$ and $g$, of the form $\sum_{n \le N} f(n) g(n+h)$, in the context of heuristic asymptotic formul\ae. Here, the integer $h\ge 0$ is called, as usual, the {\it shift}…
This paper is devoted to the study of $$ U_t(a,q):=\sum_{1\leq n_1<n_2<\cdots<n_t}\frac{q^{n_1+n_2+\cdots+n_t}}{(1+aq^{n_1}+q^{2n_1})(1+aq^{n_2}+q^{2n_2})\cdots(1+aq^{n_t}+q^{2n_t})} $$ when $a$ is one of $0, \pm 1, \pm2$. The idea builds…
All arithmetical functions $F$ satisfying Ramanujan Conjecture, i.e., $F(n)\ll_{\varepsilon}n^{\varepsilon}$, and with $Q-$smooth divisors, i.e., with Eratosthenes transform $F':=F\ast \mu$ supported in $Q-$smooth numbers, have a kind of…
We prove an interesting symmetric $q$-series identity which generalizes a result due to Ramanujan. A proof that is analytic in nature is offered, and a bijective-type proof is also outlined.
In 1918 S. Ramanujan defined a family of trigonometric sum now known as Ramanujan sums. In the last few years, Ramanujan sums have inspired the signal processing community. In this paper, we have defined an operator termed here as Ramanujan…
We consider an extension of the Ramanujan series with a variable $x$. If we let $x=x_0$, we call the resulting series: "Ramanujan series with the shift $x_0$". Then, we relate these shifted series to some $q$-series and solve the case of…
The Ramanujan $_1\psi_1$ summation theorem in studied from the perspective of $q$-Jackson integrals, $q$-difference equations and connection formulas. This is an approach which has previously been shown to yield Bailey's very-well-poised…
In a one-page fragment published with his lost notebook, Ramanujan stated two double series identities associated, respectively, with the famous Gauss Circle and Dirichlet Divisor problems. The identities contain an "extra" parameter, and…
We study a generalized class of weighted $k$-regular partitions defined by \[ \sum_{n=0}^{\infty} c_{k, r_1, r_2}(n) q^n = \prod_{n=1}^{\infty} \frac{(1 - q^{nk})^{r_1}}{(1 - q^n)^{r_2}}, \] which extends the classical $k$-regular partition…
A generalized Bailey pair, which contains several special cases considered by Bailey (\emph{Proc. London Math. Soc. (2)}, 50 (1949), 421--435), is derived and used to find a number of new Rogers-Ramanujan type identities. Consideration of…
In this paper, we state some $q$-analogues of the famous Ramanujan's Master Theorem. As applications, some values of Jackson's $q$-integrals involving $q$-special functions are computed.
We prove q-analogues of two Ramanujan-type series for $1/\pi$ from $q$-analogues of ordinary WZ pairs.
We have found several summation formulas that extend Ramanujan's psi sum. First contains a parameter $\alpha=1/N$, $N$ is a positive integer, and transforms to $q$-beta integral in the limit $N\to\infty$. The other is a $q$-analogue of…
In the present paper, we establish a general transformation for $q$-series which contains L. Wang et al's transformation involved in Nahm series. As direct applications, some concrete new transformation formulas for the ${}_{r+1}\phi_r$…
We define a new kind of classical digamma function, and establish its some fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler type sums. The main results…
We introduce the notion of the automorphic dual of a matrix algebraic group defined over $Q$. This is the part of the unitary dual that corresponds to arithmetic spectrum. Basic functorial properties of this set are derived and used both to…
We consider relationships between classical Lambert series, multiple Lambert series and classical $q$-series of the Rogers-Ramanujan type. We conclude with a contemplation on the Andrews-Dixit-Schultz-Yee conjecture.
We present here a way to evaluate a very wide class of integrals relating Ramanujans continued fraction and q-product. To do this we explore briefily a differential equation, which relates these two functions